IS IT POSSIBLE TO CHARACTERIZE THE GEOMETRY OF A REAL POROUS MEDIUM BY A DIRECT MEASUREMENT ON A FINITE SECTION? 1: THE PHASE-RETRIEVAL PROBLEM

Abstract

Macroscopic transport properties of natural porous media, such as the permeability tensor K, are the sum of uncountable microscopic events. Understanding how these microscopic events come together to yield a descriptive property, such as K, is facilitated if a set of porous descriptors Pii=1,N can be measured that synthesize all the critical microscopic properties of a real porous medium and that can be related to the computed K. The main difficulty lies in choosing the correct Pii=1,N. A description of the microgeometry of a porous medium by a set of Pii=1,N will be declared adequate if synthetic numerical porous media generated from the Pii=1,N possess the same K as the real medium. This paper is another step for creating such synthetic porous media. The candidate geometrical descriptor P1 considered herein is the autocorrelation function (ACF) measured directly on a sample of finite size v included in a porous medium of size V; V >> v. Capitalizing on the phase retrieval problem (retrieving an object from knowledge of its Fourier modulus) encountered in general imaging, it is shown that there exists a one-to-one relation between a digital thin section and its ACF. This is demonstrated using an iterative procedure, the Error Reduction/Hybrid Input Output algorithm, that allows one to recover uniquely, to within a pixel, a finite image from its ACF. A theoretical implication of this is that a direct measurement on a finite image cannot characterize the geometry of a porous medium. Yet, in stochastic modeling, quasi-infinite numerical porous media are commonly generated from acquired ACF. Such quasi-infinite stochastic porous media must include a structural noise as a practical consequence of the unicity of the relation between an image and its ACF. To correctly interpret the relation between $\new{{\em K}}$ and the microgeometry, it becomes necessary to verify that this nonimposed structural noise does not control the output K of the numerical simulations.